hvsub4 - Hilbert Space Explorer

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Theorem hvsub4 30277

Description: Hilbert vector space addition/subtraction law. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hvsub4	⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +_ℎ 𝐵) −_ℎ (𝐶 +_ℎ 𝐷)) = ((𝐴 −_ℎ 𝐶) +_ℎ (𝐵 −_ℎ 𝐷)))

**Proof of Theorem hvsub4**
Step	Hyp	Ref	Expression
1		hvaddcl 30252	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +_ℎ 𝐵) ∈ ℋ)
2		hvaddcl 30252	. . 3 ⊢ ((𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐶 +_ℎ 𝐷) ∈ ℋ)
3		hvsubval 30256	. . 3 ⊢ (((𝐴 +_ℎ 𝐵) ∈ ℋ ∧ (𝐶 +_ℎ 𝐷) ∈ ℋ) → ((𝐴 +_ℎ 𝐵) −_ℎ (𝐶 +_ℎ 𝐷)) = ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐶 +_ℎ 𝐷))))
4	1, 2, 3	syl2an 596	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +_ℎ 𝐵) −_ℎ (𝐶 +_ℎ 𝐷)) = ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐶 +_ℎ 𝐷))))
5		hvsubval 30256	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶)))
6	5	ad2ant2r 745	. . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶)))
7		hvsubval 30256	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 −_ℎ 𝐷) = (𝐵 +_ℎ (-1 ·_ℎ 𝐷)))
8	7	ad2ant2l 744	. . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐵 −_ℎ 𝐷) = (𝐵 +_ℎ (-1 ·_ℎ 𝐷)))
9	6, 8	oveq12d 7423	. . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 −_ℎ 𝐶) +_ℎ (𝐵 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))))
10		neg1cn 12322	. . . . . . 7 ⊢ -1 ∈ ℂ
11		ax-hvdistr1 30248	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (-1 ·_ℎ (𝐶 +_ℎ 𝐷)) = ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ 𝐷)))
12	10, 11	mp3an1 1448	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (-1 ·_ℎ (𝐶 +_ℎ 𝐷)) = ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ 𝐷)))
13	12	adantl 482	. . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (-1 ·_ℎ (𝐶 +_ℎ 𝐷)) = ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ 𝐷)))
14	13	oveq2d 7421	. . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐶 +_ℎ 𝐷))) = ((𝐴 +_ℎ 𝐵) +_ℎ ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ 𝐷))))
15		hvmulcl 30253	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ 𝐶) ∈ ℋ)
16	10, 15	mpan 688	. . . . . . . 8 ⊢ (𝐶 ∈ ℋ → (-1 ·_ℎ 𝐶) ∈ ℋ)
17	16	anim2i 617	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ∈ ℋ ∧ (-1 ·_ℎ 𝐶) ∈ ℋ))
18		hvmulcl 30253	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝐷 ∈ ℋ) → (-1 ·_ℎ 𝐷) ∈ ℋ)
19	10, 18	mpan 688	. . . . . . . 8 ⊢ (𝐷 ∈ ℋ → (-1 ·_ℎ 𝐷) ∈ ℋ)
20	19	anim2i 617	. . . . . . 7 ⊢ ((𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ∈ ℋ ∧ (-1 ·_ℎ 𝐷) ∈ ℋ))
21	17, 20	anim12i 613	. . . . . 6 ⊢ (((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ (𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ∈ ℋ ∧ (-1 ·_ℎ 𝐶) ∈ ℋ) ∧ (𝐵 ∈ ℋ ∧ (-1 ·_ℎ 𝐷) ∈ ℋ)))
22	21	an4s 658	. . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ∈ ℋ ∧ (-1 ·_ℎ 𝐶) ∈ ℋ) ∧ (𝐵 ∈ ℋ ∧ (-1 ·_ℎ 𝐷) ∈ ℋ)))
23		hvadd4 30276	. . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ (-1 ·_ℎ 𝐶) ∈ ℋ) ∧ (𝐵 ∈ ℋ ∧ (-1 ·_ℎ 𝐷) ∈ ℋ)) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))) = ((𝐴 +_ℎ 𝐵) +_ℎ ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ 𝐷))))
24	22, 23	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))) = ((𝐴 +_ℎ 𝐵) +_ℎ ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ 𝐷))))
25	14, 24	eqtr4d 2775	. . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐶 +_ℎ 𝐷))) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))))
26	9, 25	eqtr4d 2775	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 −_ℎ 𝐶) +_ℎ (𝐵 −_ℎ 𝐷)) = ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐶 +_ℎ 𝐷))))
27	4, 26	eqtr4d 2775	1 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +_ℎ 𝐵) −_ℎ (𝐶 +_ℎ 𝐷)) = ((𝐴 −_ℎ 𝐶) +_ℎ (𝐵 −_ℎ 𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 (class class class)co 7405 ℂcc 11104 1c1 11107 -cneg 11441 ℋchba 30159 +_ℎ cva 30160 ·_ℎ csm 30161 −_ℎ cmv 30165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-hfvadd 30240 ax-hvcom 30241 ax-hvass 30242 ax-hfvmul 30245 ax-hvdistr1 30248

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-sub 11442 df-neg 11443 df-hvsub 30211

This theorem is referenced by: hvaddsub4 30318 5oalem2 30895 3oalem2 30903

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